Note that \(x,Q\)  is a regular sequence and hence \(e(S/\mathfrak {p})=2\) . Therefore \(m(\mathfrak {p})\le 2\) . Hence \(m(\mathfrak {p})=2\) , and we must have that \(\mathfrak {p}\)  is the only minimal prime and \(I\)  is \(\mathfrak {p}\) -primary. Consider the chain of \(S_{\mathfrak {p}}\) -submodules given by \((0)\subset (x^2)\subsetneq (x)\subsetneq (S/I)_{\mathfrak {p}}\) . Since \(m(\mathfrak {p})=\mathrm {length}(S/I)_{\mathfrak {p}})=2\) , we must have \(x^2=0\)  in \((S/I)_{\mathfrak {p}}\) . Therefore by [1]}, we conclude that \(x^2\in I\) . Since \(Q_1,Q_2\)  are of degree 2, we must have that \(x^2\in \mathrm {span}(Q_1,Q_2)\) .
In this case, 3.(b) occurs.
